The near pressure field of co-axial subsonic jets
TL;DR: In this paper, an analysis of the axial, temporal and azimuthal structure of the pressure field of a co-axial jet with and without serrations on the secondary nozzle lip is presented.
Abstract: Results are presented from pressure measurements performed in the irrotational near field of unbounded co-axial jets. Measurements were made for a variety of velocity and temperature ratios, and configurations both with and without serrations on the secondary nozzle lip. The principal objective of the study is to better understand the near pressure field of the jet, what it can tell us regarding the underlying turbulence structure, and in particular how it can be related to the source mechanisms of the flow.A preliminary analysis of the axial, temporal and azimuthal structure of the pressure field shows it to be highly organized, with axial spatial modes (obtained by proper orthogonal decomposition) which resemble Fourier modes. The effects of serrations on the pressure fluctuations comprise a global reduction in level, a change in the axial energy distribution, and a modification of the evolution of the characteristic time scales.A further analysis in frequency–wavenumber space is then performed, and a filtering operation is used to separate the convective and propagative footprints of the pressure field. This operation reveals two distinct signatures in the propagating component of the field: a low-frequency component which radiates at small angles to the flow axis and is characterized by extensive axial coherence, and a less-coherent high-frequency component which primarily radiates in sideline directions. The serrations are found to reduce the energy of the axially coherent propagating component, but its structure remains fundamentally unchanged; the high-frequency component is found to be enhanced. A further effect of the serrations involves a relative increase of the mean-square pressure level of the acoustic component – integrated over the measurement domain – with respect to the hydrodynamic component. The effect of increasing the velocity and temperature of the primary jet involves a relative increase in the acoustic component of the near field, while the hydrodynamic component remains relatively unchanged: this shows that the additional acoustic energy is generated by the mixing region which is produced by the interaction of the inner and the outer shear layers, whereas the hydrodynamic component of the near field is primarily driven by the outer shear layer.
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TL;DR: In this article, the spectral proper orthogonal decomposition (SPOD) has been studied in the context of the analysis of the Ginzburg-Landau equation and a turbulent jet.
Abstract: We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.
756 citations
Cites background from "The near pressure field of co-axial..."
...There are some articles that treat the two variants as separate methods, but the relationship between them is not explored in detail (Aubry et al., 1988; Picard & Delville, 2000; Chen & Kareem, 2005; Tinney & Jordan, 2008; Holmes et al., 2012; Taira et al., 2017)....
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TL;DR: In this paper, the authors investigated the extent to which pressure and velocity fluctuations in subsonic, turbulent round jets can be described as linear perturbations to the mean flow field.
Abstract: Previous work has shown that aspects of the evolution of large-scale structures, particularly in forced and transitional mixing layers and jets, can be described by linear and nonlinear stability theories. However, questions persist as to the choice of the basic (steady) flow field to perturb, and the extent to which disturbances in natural (unforced), initially turbulent jets may be modelled with the theory. For unforced jets, identification is made difficult by the lack of a phase reference that would permit a portion of the signal associated with the instability wave to be isolated from other, uncorrelated fluctuations. In this paper, we investigate the extent to which pressure and velocity fluctuations in subsonic, turbulent round jets can be described as linear perturbations to the mean flow field. The disturbances are expanded about the experimentally measured jet mean flow field, and evolved using linear parabolized stability equations (PSE) that account, in an approximate way, for the weakly non-parallel jet mean flow field. We utilize data from an extensive microphone array that measures pressure fluctuations just outside the jet shear layer to show that, up to an unknown initial disturbance spectrum, the phase, wavelength, and amplitude envelope of convecting wavepackets agree well with PSE solutions at frequencies and azimuthal wavenumbers that can be accurately measured with the array. We next apply the proper orthogonal decomposition to near-field velocity fluctuations measured with particle image velocimetry, and show that the structure of the most energetic modes is also similar to eigenfunctions from the linear theory. Importantly, the amplitudes of the modes inferred from the velocity fluctuations are in reasonable agreement with those identified from the microphone array. The results therefore suggest that, to predict, with reasonable accuracy, the evolution of the largest-scale structures that comprise the most energetic portion of the turbulent spectrum of natural jets, nonlinear effects need only be indirectly accounted for by considering perturbations to the mean turbulent flow field, while neglecting any non-zero frequency disturbance interactions.
265 citations
TL;DR: In this article, the velocity field of unforced, high Reynolds number, subsonic jets, issuing from round nozzles with turbulent boundary layers, is measured using a hot-wire anemometer and a stereoscopic, time-resolved PIV system.
Abstract: We study the velocity fields of unforced, high Reynolds number, subsonic jets, issuing from round nozzles with turbulent boundary layers. The objective of the study is to educe wavepackets in such flows and to explore their relationship with the radiated sound. The velocity field is measured using a hot-wire anemometer and a stereoscopic, time-resolved PIV system. The field can be decomposed into frequency and azimuthal Fourier modes. The low-angle sound radiation is measured synchronously with a microphone ring array. Consistent with previous observations, the azimuthal wavenumber spectra of the velocity and acoustic pressure fields are distinct. The velocity spectrum of the initial mixing layer exhibits a peak at azimuthal wavenumbers ranging from 4 to 11, and the peak is found to scale with the local momentum thickness of the mixing layer. The acoustic pressure field is, on the other hand, predominantly axisymmetric, suggesting an increased relative acoustic efficiency of the axisymmetric mode of the velocity field, a characteristic that can be shown theoretically to be caused by the radial compactness of the sound source. This is confirmed by significant correlations, as high as 10 %, between the axisymmetric modes of the velocity and acoustic pressure fields, these values being significantly higher than those reported for two-point flow–acoustic correlations in subsonic jets. The axisymmetric and first helical modes of the velocity field are then compared with solutions of linear parabolized stability equations (PSE) to ascertain if these modes correspond to linear wavepackets. For all but the lowest frequencies close agreement is obtained for the spatial amplification, up to the end of the potential core. The radial shapes of the linear PSE solutions also agree with the experimental results over the same region. The results suggests that, despite the broadband character of the turbulence, the evolution of Strouhal numbers 0.3 ≤ St ≤ 0.9 and azimuthal modes 0 and 1 can be modelled as linear wavepackets, and these are associated with the sound radiated to low polar angles.
226 citations
Cites methods from "The near pressure field of co-axial..."
...Measurements using line arrays of microphones in the near field reveal a hydrodynamic wave extending several jet diameters downstream of the nozzle exit (Picard & Delville 2000; Tinney & Jordan 2008)....
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TL;DR: In this paper, an azimuthal ring array of six microphones, whose polar angle, θ, was progressively varied, allows the decomposition of the acoustic pressure into axisymmetric Fourier modes.
Abstract: We present experimental results for the acoustic field of jets with Mach numbers
between 0.35 and 0.6. An azimuthal ring array of six microphones, whose polar
angle, θ, was progressively varied, allows the decomposition of the acoustic pressure
into azimuthal Fourier modes. In agreement with past observations, the sound field for
low polar angles (measured with respect to the jet axis) is found to be dominated by
the axisymmetric mode, particularly at the peak Strouhal number. The axisymmetric
mode of the acoustic field can be clearly associated with an axially non-compact
source, in the form of a wavepacket: the sound pressure level for peak frequencies is
found be superdirective for all Mach numbers considered, with exponential decay as a
function of (1 – M_c cos θ)^2, where M_c is the Mach number based on the phase velocity
U_c of the convected wave. While the mode m = 1 spectrum scales with Strouhal
number, suggesting that its energy content is associated with turbulence scales,
the axisymmetric mode scales with Helmholtz number – the ratio between source
length scale and acoustic wavelength. The axisymmetric radiation has a stronger
velocity dependence than the higher-order azimuthal modes, again in agreement with
predictions of wavepacket models. We estimate the axial extent of the source of
the axisymmetric component of the sound field to be of the order of six to eight
jet diameters. This estimate is obtained in two different ways, using, respectively,
the directivity shape and the velocity exponent of the sound radiation. The analysis
furthermore shows that compressibility plays a significant role in the wavepacket
dynamics, even at this low Mach number. Velocity fluctuations on the jet centreline
are reduced as the Mach number is increased, an effect that must be accounted for
in order to obtain a correct estimation of the velocity dependence of sound radiation.
Finally, the higher-order azimuthal modes of the sound field are considered, and a
model for the low-angle sound radiation by helical wavepackets is developed. The
measured sound for azimuthal modes 1 and 2 at low Strouhal numbers is seen to
correspond closely to the predicted directivity shapes.
195 citations
TL;DR: In this paper, three simplified wave-packet models of the coherent structures in subsonic jets are presented, and the dependence of the radiated sound on the temporal variations of the amplitude and spatial extent of the modulations are studied separately in the first two model problems, being considered together in the third.
180 citations
References
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TL;DR: The Navier-Stokes equations are well-known to be a good model for turbulence as discussed by the authors, and the results of well over a century of increasingly sophisticated experiments are available at our disposal.
Abstract: It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical
3,721 citations
TL;DR: In this article, the wall region of a turbulent boundary layer is modelled by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem.
Abstract: We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier–Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system.
1,271 citations
"The near pressure field of co-axial..." refers background or methods in this paper
...The application of this technique has been well documented (Glauser & George 1987; Aubry et al. 1988; Berkooz, Holmes & Lumley 1993; Delville et al. 1999)....
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...By truncation of the convergent series (i.e. m < N), a low-order approximation of the kernel is obtained which can provide insight into the spatial structure of the flow features which are best correlated with the energy-integrable field (Aubry et al. 1988)....
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TL;DR: In this article, the mechanisms of sound generation in a Mach 0.9, Reynolds number 3600 turbulent jet are investigated by direct numerical simulation and the results show that the phase velocities of significant components range from approximately 5% to 50% of the ambient sound speed.
Abstract: The mechanisms of sound generation in a Mach 0.9, Reynolds number 3600 turbulent jet are investigated by direct numerical simulation. Details of the numerical method are briefly outlined and results are validated against an experiment at the same flow conditions (Stromberg, McLaughlin & Troutt 1980). Lighthill's theory is used to define a nominal acoustic source in the jet, and a numerical solution of Lighthill's equation is compared to the simulation to verify the computational procedures. The acoustic source is Fourier transformed in the axial coordinate and time and then filtered in order to identify and separate components capable of radiating to the far field. This procedure indicates that the peak radiating component of the source is coincident with neither the peak of the full unfiltered source nor that of the turbulent kinetic energy. The phase velocities of significant components range from approximately 5% to 50% of the ambient sound speed which calls into question the commonly made assumption that the noise sources convect at a single velocity. Space–time correlations demonstrate that the sources are not acoustically compact in the streamwise direction and that the portion of the source that radiates at angles greater than 45° is stationary. Filtering non-radiating wavenumber components of the source at single frequencies reveals that a simple modulated wave forms for the source, as might be predicted by linear stability analysis. At small angles from the jet axis the noise from these modes is highly directional, better described by an exponential than a standard Doppler factor.
632 citations
01 Jan 1996
TL;DR: In this paper, two similarity spectra, one for the noise from the large turbulence structures/instability waves of the jet flow, the other for the fine-scale turbulence, are identified.
Abstract: It is argued that because of the lack of intrinsic length and time scales in the core part of the jet flow, the radiated noise spectrum of a high-speed jet should exhibit similarity. A careful analysis of all the axisymmetric supersonic jet noise spectra in the data-bank of the Jet Noise Laboratory of the NASA Langley Research Center has been carried out. Two similarity spectra, one for the noise from the large turbulence structures/instability waves of the jet flow, the other for the noise from the fine-scale turbulence, are identified. The two similarity spectra appear to be universal spectra for axisymmetric jets. They fit all the measured data including those from subsonic jets. Experimental evidence are presented showing that regardless of whether a jet is supersonic or subsonic the noise characteristics and generation mechanisms are the same. There is large turbulence structures/instability waves noise from subsonic jets. This noise component can be seen prominently inside the cone of silence of the fine-scale turbulence noise near the jet axis. For imperfectly expanded supersonic jets, a shock cell structure is formed inside the jet plume. Measured spectra are provided to demonstrate that the presence of a shock cell structure has little effect on the radiated turbulent mixing noise. The shape of the noise spectrum as well as the noise intensity remain practically the same as those of a fully expanded jet. However, for jets undergoing strong screeching, there is broadband noise amplification for both turbulent mixing noise components. It is discovered through a pilot study of the noise spectrum of rectangular and elliptic supersonic jets that the turbulent mixing noise of these jets is also made up of the same two noise components found in axisymmetric jets. The spectrum of each individual noise component also fits the corresponding similarity spectrum of axisymmetric jets.
459 citations
"The near pressure field of co-axial..." refers result in this paper
...These observations are consistent with the two similarity spectra proposed by Tam et al. (1996); however, the dynamics of the jet responsible for each of these signatures remain poorly understood....
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